Social Theory introduction up to Arrow's and Gibbard–Satterthwaite's Theorems. Have fun.

1. Social Choice Theory
Enrico Franchi
efranchi@ce.unipr.it

2. Group Decisions
} Agents are required to choose among a set of outcomes
Ω = {ω1 , ω2 ,…}
} Agents can choose one outcome in Ω
} Agents can express a preference of outcomes
} Let Π ( Ω ) the set of preference orderings of outcomes
} We also write ω 1 i ω 2 to express that agent i prefers ω1
to ω2
2 Enrico Franchi (efranchi@ce.unipr.it)

3. Social Welfare
} A social welfare function takes the voter preferences and
produces a social preference order:
f : Π (Ω) → Π (Ω)
N
or in the slightly simplified form:
f : Π (Ω) → Ω
N
} We write ω 1 * ω 2 to express that the first outcome
ranked above the second in the social outcome
3 Enrico Franchi (efranchi@ce.unipr.it)

4. Plurality
} Simplest voting procedure: used to select a single
outcome (candidate)
} Everyone submits his preference order, we count how
many times each candidate was ranked first
} Easy to implement and to understand
} If the outcomes are just 2, it is called simple majority voting
} If they are more than two, problems arise
4 Enrico Franchi (efranchi@ce.unipr.it)

5. Voting in the UK
} Three main parties: Voters
Labour Party (left-wing)
}
} Liberal Democrats (center- Conser
vative Labour
left) Party
Party
} Conservative Party (right- 44% 44%
wing)
} Left-wing voter: ω L  ω D  ω C
} Center voter:ω D  ω L  ω C
} Right-wing voter:ω C  ω D  ω L Liberal
Democ
} Tactical Voting rats
} Strategic Manipulation 12%
5 Enrico Franchi (efranchi@ce.unipr.it)

6. Condorcet’s Paradox
} Consider this election:
Ω = {ω 1 ,ω 2 ,ω 3 } Ag = {1,2,3}
ω 1 1 ω 2 1 ω 3
ω 3 2 ω 1 2 ω 2
ω 2 3 ω 3 3 ω 1
} No matters the outcome we choose: two thirds of the
electors will be unhappy
6 Enrico Franchi (efranchi@ce.unipr.it)

7. Sequential Majority
} Series of pair-wise elections, the winner will go on to the
next election
} An agenda is the strategy we choose to order the
elections (linear, binary tree)
} An outcome is a possible winner if there is some agenda
which would make that outcome the overall winner
} An outcome is a Condorcet winner if it is the overall
winner for every possible agenda
} Can we choose the agenda to choose a winner?
7 Enrico Franchi (efranchi@ce.unipr.it)

8. Borda Count and Slater Ranking
} Borda Count
} We have K outcomes
} Each time an outcome is in the j-th position for some agent, we
increment its counter by K-j
} We order the outcomes according to their counter
} Good for single candidates
} Slater ranking
} Tries to be as close to the majority graph as possible
} Unfortunately, is NP-hard
8 Enrico Franchi (efranchi@ce.unipr.it)

9. Properties
} Pareto condition: if every agent ranks ωi above ωj,
then ω i * ω j
} Plurality, Borda
} Condorcet winner: if an outcome is a Condorcet
winner, then it should be ranked first
} Sequential majority elections
} Independence of Irrelevant Alternatives (IIA): social
ranking of two outcomes should only be affected by the
way that they are ranked in their preference orders
} Almost no protocol satisfies IIA
9 Enrico Franchi (efranchi@ce.unipr.it)

10. Properties
} Dictatorship: a social welfare function f is a dictatorship
if for some voter j we have that:
f (ω1 ,…, ωN ) = ω j
} Unrestricted Domain: for any set of individual voter
preferences, the social welfare function should yield a
unique and complete ranking of societal choices.
} E.g., not random, always answers, does not “loop”
10 Enrico Franchi (efranchi@ce.unipr.it)

11. Arrow’s Theorem
} There is no voting procedure for elections with more
than two outcomes that satisfies
} Non-dictatorship
} Unrestricted Domain
} Pareto
} Independence of Irrelevant Alternatives
11 Enrico Franchi (efranchi@ce.unipr.it)

12. Gibbard-Satterthwaite’s Theorem
} Sometimes voters “lie” in order to obtain a better
outcome
} Is it possible to devise a voting procedure that is not
subject to such manipulation?
} Manipulation (i prefers ωi):
f (ω 1 ,…,ω i '…,ω n ) i f (ω 1 ,…,ω i …,ω n )
} The only procedure that cannot be manipulated and
satisfies the Pareto condition is dictatorship
12 Enrico Franchi (efranchi@ce.unipr.it)

13. Complexity and Manipulation
} Even if all procedures can be manipulated, can we devise
procedures which are hard to manipulate?
} Hard means “difficult to compute” in an algorithmic
sense, e.g., NP-Hard procedures
} These procedures are easy (polynomial) to compute?
} Second-order Copeland may be “difficult” to manipulate
} In theory it is NP-Hard
} However, it is only a worst case complexity
13 Enrico Franchi (efranchi@ce.unipr.it)

14. References
1. Multiagent Systems: Algorithmic, Game-Theoretic, and
Logical Foundations; Yoav Shoham and Kevin Leyton-
Brown; Cambridge Press
2. Game Theory: Analysis of Conflict; Roger B. Myerson;
Harvard Press
3. An Introduction to Multi-Agent Systems; Michael
Wooldridge; Wiley Press
14 Enrico Franchi (efranchi@ce.unipr.it)